bibkey: oeis2024a376317
authors: Mikhail Kurkov
year: 2024
title: OEIS A376317, periodic shifts of a two-index recurrence
url: https://oeis.org/A376317
doi: null
claim: "The shifts d mod 2 and the indicator of d mod 3 being positive both give A006318."
strata_touched:
  - D5/S1/Recurrence/Algebraic/PeriodicBoundarySchroder
license: citation-only
triage: anchor

Periodic shifts and the large Schroder diagonal

Kurkov’s original natural array has P(0,m)=1 for every m. For positive n, P(n,m)=P(n-1,m) when m<=n; otherwise P(n,m)=P(n,m-1)+P(n-1,m+f(m-n)). The zeroth-row rule takes priority. The conjectured sequence is P(n+1,n+1), beginning at n=0, with both f(d)=d mod 2 and f(d)=0 if 3 divides d, 1 otherwise.

Verified locator

The original source is https://oeis.org/A376317/internal, revision 25. The conjecture was introduced in revision 15 and is attributed to Mikhail Kurkov in 2024. Its exact wording is: “Conjecture: cases f(n) = n mod 2 and f(n) = [(n mod 3) > 0] both gives A006318.” The entry’s offset is 0,2. A376388 repeats the same conjecture and is not a second independent question. The main sequences in those entries use different shifts; those main sequences are not the present target. OEIS source text is licensed CC-BY-SA-4.0; the quotation is attributed to Kurkov and linked to its source. The Lean proof and mathematical exposition are repository derivations. Mathlib’s large Schroder series and its quadratic equation are reused under Mathlib’s Apache-2.0 license.

Search boundary

The source was retrieved successfully at 2026-10-28T20:32:44Z and still labels this exact conjunction a conjecture. The literature window includes arXiv metadata queries for A376317, A376388, and Kurkov with Schroder; complete pinned indexes of 24 LeanOpenProblems-results runs, including 19 OEIS runs and 3468 OEIS sample directories mapped to 492 conjecture records; and targeted third-party Lean queries with ten distinct relevant Lean blobs inspected. No exact solution was identified in that window. The published proof artifacts were not exhaustively searched for semantic equivalence. Broad code-search queries were capped at their first 100 results; rate limits and a LeanSearch service gap remain. These findings do not establish worldwide priority or exhaustive absence of prior proofs.